In a small saucepan over medium-high, combine the rice and water. Bring to a boil, then reduce the heat to a simmer. Cook, covered, for 30 minutes. Let stand for a few minutes, then pour through a mesh strainer to discard any excess water. Return the rice to the pot over medium-high heat. Add 2 cups of the milk, the sugar, the cinnamon stick and a hefty pinch of salt. Bring to a boil, then reduce the heat to simmer and cook, uncovered and stirring occasionally, for 40 minutes.

In a small bowl, beat the eggs with the remaining ½ cup milk. Whisk in a large spoonful of the hot rice mixture. Add the egg mixture to the rice and cook over medium-low heat, stirring constantly, until the custard coats the back of a spoon, 4 to 5 minutes. Do not let the rice pudding boil or the eggs will scramble.

Remove the saucepan from the heat, stir in the vanilla and transfer the rice pudding to a bowl. Cover the pudding and chill until cold, at least 2 hours. The pudding will thicken as it chills. To serve, discard the cinnamon stick and divide the rice pudding among 4 bowls. Top each portion with some of the ginger.